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Hsuan et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng)
in press
Journal of Zhejiang University-SCIENCE A (Applied Physics & Engineering)
ISSN 1673-565X (Print); ISSN 1862-1775 (Online)
www.zju.edu.cn/jzus; www.springerlink.com
E-mail: [email protected]
Finite element modeling of superplastic co-doped
yttria-stabilized tetragonal-zirconia polycrystals
Hsuan-Teh HU†1, Shih-Tsung TSENG1, Alice HU2
d
(1Department of Civil Engineering, National Cheng Kung University, Tainan, China)
2
( Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong, China)
†
E-mail: [email protected]
ite
Received May. 30, 2015; Revision accepted May. 10, 2016; Crosschecked
ed
Abstract: Y2O3-stabilized tetragonal-zirconia polycrystals (Y-TZP) have been shown to have superplastic properties at high
temperatures, opening a way for the manufacture of complex pieces for industrial applications by a variety of techniques.
However, before that is possible, it is important to analyze the deformation and fracture mechanisms at a macroscopic level based
on continuum theory. In this paper, an elastic-plastic material model with a theoretical large deformation is constructed to simulate
the true stress - true strain relationships of superplastic ceramics. The simplified constitutive law used for the numerical
simulations is based on piecewise linear connections at the turning points of different deformation stages on the experimental
stress-strain curves. The finite element model (FEM) is applied to selected tensile tests on 3-mol%-Y-TZP (3Y-TZP) co-doped
with germanium oxide and other oxides (titanium, magnesium, and calcium) to verify its applicability. The results show that the
stress-strain characteristics and the final deformed shapes in the finite element analysis (FEA) agree well with the tensile test
experiments. It can be seen that the FEM presented can simulate the mechanical behavior of superplastic co-doped 3Y-TZP
ceramics and that it offers a selective numerical simulation method for advanced development of superplastic ceramics.
Key words: Finite element enalysis, Y-TZP, Superplasticity
doi:10.1631/jzus.A1500159
Document code: A
un
1. Introduction
Ceramics have several excellent properties that
are unique among materials. They are extremely hard,
have good resistance to fatigue, abrasion, and
chemical corrosion, and are resistant to erosion at
high temperatures. Hence, ceramics can be used in
extreme environments, and their applications are
extensive in both household and industrial products.
However, they have a fatal weakness, brittleness,
which means that they break at small levels of strain
and cannot withstand extensive elongation under load.
Thus, there has been a significant research effort to
reduce the brittleness of ceramics.
© Zhejiang University and Springer-Verlag Berlin Heidelberg 2016
1
CLC number:
Through the tailoring of very refined particles,
tetragonal-zirconia polycrystals (TZP) retain a
metastable tetragonal phase at room temperature and
have been shown to have excellent plastic properties
(Garvie et al., 1975). Wakai et al. (1986)
demonstrated that a 3Y-TZP with an average grain
size smaller than 0.3 μm could be elongated
superplastically to greater than 120% of the nominal
strain under conditions of constant strain rates
ranging from 1.1×10-4s-1 to 5.5×10-4s-1 at a
temperature of 1450 °C. It was that research that first
revealed the superplasticity of 3Y-TZP. Since then,
much effort has been devoted to the development of
ceramics with improved superplasticity. JiménezMelendo et al. (1998) analyzed extensive
experimental data for fine-grained Y2O3-stabilized
Husan et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng)
d
published. However, most of this research has been
focused on metallic materials such as titanium-based,
aluminum-based, zinc-based, copper-based, and
magnesium-based alloys. These FE models have
included combinations of several simulation
conditions, which can generally be classified into
different dimensions (2-D or 3-D) with different
constitutive models (elastic-plastic, rigid-plastic,
elastic-viscoplastic, or rigid-viscoplastic); some of
these models have included the microstructure
evolution of grain growth or void growth to regulate
the stress-strain relationships of various materials,
different elements, and the assumption of isotropic or
anisotropic material properties, among other
considerations. For example, Hsu and Chu (1995),
Wang and Budiansky (1978) used membrane
elements, and Hsu and Shien (1997) used shell
elements in a two-dimensional condition to simulate
the sheet metal stamping process; in addition, their
FEM contained an elastic-plastic constitutive model
for which post-yield anisotropic material behavior
was formulated by the flow rule associated with Hill’s
quadratic yield criterion. Lee and Kobayash (1973)
simulated superplastic metal formation problems by
using two-dimensional shell elements with a
rigid-plastic constitutive model which was related to
the associated flow rule with the von-Mises quadratic
yield criterion for isotropic materials, or with Hill’s in
the case of anisotropic materials. On the other hand,
El-Morsy et al. (2001), Giuliano (2005, 2006),
Zienkiewicz and Godbole (1974), Liew et al., (2003),
and Kim et al. (1996) simulated superplastic alloys as
non-Newtonian viscous flow materials with a
rigid-viscoplastic constitutive model to study SPF
problems. Among these, Liew et al. (2003) combined
both grain growth and void growth evolutions within
the model to incorporate the effect of the strain rate.
On the other hand, Kim et al. (1996) combined a
strain rate controlling function in their model that
could control the pressure to maintain the maximum
strain rate near a target value during analysis and
obtain the optimal pressure-time relationships for the
SPF process under the prerequisite that the material
did not crack. Similarly, for research on SPF problems
that simulate superplastic alloys as non-Newtonian
viscous flow materials using an elastic-viscoplastic
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ite
ZrO2 polycrystals (Y-SZP) containing 2-4 mol%
Y2O3. The analyses were performed as a function of
stress, grain size, and material purity at temperatures
between 1250 °C and 1450 °C. It was concluded that
for high-purity Y-SZP (impurity content less than 10
wt%), the constitutive equation for superplastic flow
in region II (the high-stress region) was similar to that
for superplastic metallic materials. On the other hand,
it was determined that the equation for region I (the
low-stress region) should be formulated by
incorporating a threshold stress approach. However,
the stress-strain relationships for low-purity Y-SZP
(impurity content greater than 0.10 wt%) were
different from those of metallic systems
(Jiménez-Melendo et al., 1998, 1999). It was not
possible to model the behavior of ceramics with a
highly viscous liquid phase by using the metallic
constitutive creep models relevant to a steady-state
strain rate at high temperatures (Bieler et al., 1996).
Wakai et al. (1999) also noted that there existed no
model that could explain all the experimental data on
Y-TZP deformation mechanisms. Currently, in a deep
analysis of the multi-scale (macroscopic) nature of
ceramic superplasticity, one of the major goals that
has to be fulfilled is correlation between the behavior
of one individual grain under shear and normal
stresses and their average collective behavior
(Domínguez-Rodríguez and Gómez-García, 2010).
Combining the superplastic properties of
ceramics with processing techniques, such as sheet
formation, blowing, stamping, forging, and spinning,
would support the industrial manufacture of complex
ceramic pieces for applications in aerospace, defense,
and automobile manufacturing, among others
(Domínguez-Rodríguez and Gómez-García, 2010).
However, before that is practical, we must gain more
understanding of superplastic ceramics in respect of
mechanical stress distribution and fracture
mechanisms at a macroscopic level when the
materials deform. So here we intend, by means of
solid mechanics or plasticity theory, to provide further
analysis of the deformation progress of superplastic
ceramics so that uses of this kind of material may be
developed on the basis of safe designs.
A great deal of research on the FEA of
superplastic forming (SPF) processes has been
in press
2
d
germanium oxide, titanium oxide, calcium oxide, and
magnesium oxide in the following compositions: 2
mol% Ge-1 mol% Mg (2Ge-1Mg); 2 mol% Ge-1
mol% Ca (2Ge-1Ca); and 2 mol% Ge-2 mol% Ti
(2Ge-2Ti). The mixed powders were manufactured
into specimens of a theoretical density greater than
95% through processes of ball milling, drying,
sieving, pressing, and sintering. Then the specimens
were trimmed into the shape of a dog bone for the
tensile test samples, whose middle gauge region had
an approximate cross section of 2 mm × 2 mm and a
length of 13.4 mm. The tensile tests were conducted
at a temperature of 1400 °C under a quasi-static strain
control condition. The stress-strain characteristics of
the experimental results are shown in Fig. 1, and the
corresponding final deformed shapes are depicted in
Fig. 2.
un
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ite
constitutive model, Chandra (1988) and Nazzal et al.
(2011) incorporated grain growth evolution;
Abu-Farha and Khraisheh (2007) incorporated both
grain and void growth evolutions; Chen et al. (2001),
Yenihayat et al. (2005), Hassan et al. (2003), and Li et
al. (2004) combined the strain rate controlling
capability, and Tao and Keavey (2004), Nazzal and
Khraisheh (2008), Ding et al., (1995), Huh et al.
(1995), Nazzal et al. (2004), and Lin (2003) joined
microstructure evolutions and the strain rate
controlling capability together within their models.
To date, FEA on superplastic problems has been
done almost exclusively for metallic alloys, and there
has been comparatively little FEA on superplastic
ceramics. Since the basic properties of chemical
bonding in metals and ceramics are different, the
superplastic deformation mechanisms of these two
kinds of materials are not the same. A total
stress-strain history of superplastic deformation
includes the low-stress and high-stress regions of the
sigmoidal creep behavior as represented by a double
logarithmic strain rate vs. stress axes, and therefore
the constitutive characteristics of superplastic metals
and ceramics are different from each other. Hence, a
precise numerical simulation of a complete
stress-strain progress is very important for subsequent
mechanically-based analysis, for example, the
simulation of fracture mechanisms and the
development of safe design specifications. The aim of
this paper is to construct a numerical FEM to simulate
the stress-strain progress of co-doped 3Y-TZP
ceramics based on the continuum theory of plasticity.
The constitutive model here embedded in the FEM is
an elastic-plastic model that formulates the elastic
behavior using Hooke’s law and the subsequent work
hardening behavior by combining the associated
von-Mises flow rule with the isotropic hardening rule.
in press
Fig. 1 Stress-strain characteristics of tensile tests for
3Y-TZP and three co-doped materials [After Sasaki et al.,
2001]
2. Geometry of Experimental Samples
The presented FEM was verified with the tensile
test experiments performed by Sasaki et al. (2001) on
co-doped 3Y-TZP ceramics. In the experiments,
commercially available high-purity 3Y-TZP powders
were used as the base material, and the dopants were
3
Fig. 2 Photograph of deformed tensile samples for 3Y-TZP
and three co-doped materials [After Sasaki et al., 2001]
The experiments showed the co-doped 3Y-TZP
had excellent superplastic properties, with the largest
3. Constitutive Equations
Equation (5) is called the associated flow rule
because the plastic flow is associated with the yield
criterion. The loading surface, which defines the
boundary of the current elastic region, is the
subsequent yield surface for an elastoplastically
deformed material under combined states of stress,
and it can generally be written as
f(σ ij ,εijp ,k)=F(σ ij ,εijp )  k 2 (ε p )=0
(6)
ite
In this paper, we assumed the superplastic
co-doped 3Y-TZP ceramics to be homogeneous and
isotropic. The chosen constitutive model embedded in
the present FEM was an elastic-plastic model. The
elastic behavior was formulated using Hooke’s law,
and the work hardening plastic stress-strain
relationships were derived by combining the flow rule
associated with the von-Mises quadratic yield
criterion with the isotropic hardening rule. According
to the theory of plasticity, the relative equations of the
elastic-plastic model can be derived as follows (Chen
and Han, 2007).
The total strain increment is decomposed into
two parts,
d ij  d ije  d ijp
(1)
occur, and g is known as the plastic potential function.
The equation g(σ ij ,εijp ,k)=constant defines a surface
of plastic potential in a nine-dimensional stress space.
When the yield function and the plastic potential
function coincide, f=g, and then the plastic flow
equations can be expressed as follows:
f
(5)
dεijp =dλ
σ ij
d
nominal strain of 988% in the 2Ge-2Ti composition.
Furthermore, as indicated from an inspection using
scanning electron microscopy (SEM), all samples
consisted of a homogeneous and equiaxed
microstructure, and no amorphous phases were found
(Sasaki et al., 2001). Hence, they could be regarded as
homogeneous and isotropic materials at the
macroscopic scale. The average grain size of the four
compositions was: 3Y-TZP, 0.38µm; 2Ge-1Mg,
0.44µm; 2Ge-1Ca, 0.55µm; 2Ge-2Ti, 0.51µm.
in press
ed
The size of the yield surface is governed by the
hardening parameter k2 expressed as a function of ε p ,
called the effective strain. Hence, the parameter k2
depends upon the plastic strain history. The function
F(σ ij ,εijp ) defines the shape of the yield surface.
Moreover, for a work-hardening material, the
hardening rule is necessary to describe the rule for the
evolution of the loading surface. Since we assumed
the analyzed material to be isotropic, we took the von
Mises yield function as the plastic potential and the
isotropic hardening rule, which expanded the initial
yield surface uniformly without distortion and
translation, as the evolution of the loading surface.
When the von Mises yield criterion is used, we obtain
(7)
F(σij ,εijp )=J 2
un
where the elastic strain increment, dije , is related to
the stress increment, dσ ij , by Hooke’s law as
dσ ij =Cijkl dεkle
(2)
while Cijkl is the tensor of elastic modulus. For a
linear-elastic isotropic material, Cijkl can be expressed
in terms of the two elastic constants: shear modulus,
G, and Poisson’s ratio, ν:
ν


(3)
Cijkl =2G  δik δ jl +
δij δkl 
1-2ν


The plastic strain increment, dijp , can be
generally expressed by a non-associated flow rule in
the form
g
(4)
dεijp =dλ
σ ij
where dλ is a positive scalar factor of proportionality,
which is non-zero only when plastic deformations
4
with J 2 expressed in the following as the invariant of
the stress deviator tensor:
1
(8)
J 2 = sij sij
2
Then Eq. (6) becomes
1
(9)
f(σ ij ,εijp ,k)= sij sij  k 2 (ε p )=0
2
where sij denotes the stress deviator tensor defined by
subtracting the spherical state of stress from the actual
state of stress.
1
(10)
sij =σ ij  σ kk δij
3
For practical use of the work-hardening theory
of plasticity, the hardening parameters in the loading
From the above equations, when the plastic flow
occurs, the constitutive law of stress-strain
relationships can be derived as follows:
dσ ij =Cijkl dεkle =Cijkl (d  kl -dεklp )
=Cijkl (dεkl -dλ
f
)
σ kl


1 f
f
=Cijkl  d  kl 
Cmnst
d  st 
σ kl
h σ mn



1 f
f 
=Cijkl  δsk δtl 
Cmnst
 dεst
h σ mn
σ kl 


1 f
f 
=  Cijst 
Cmnst Cijkl
 dεst
h σ mn
σ kl 

ite
and for the uniaxial test, σe=σ1 and σ2=σ3=0.
Therefore n=2, C=1/3 and
3
(12)
σ e = 3J 2 =
sij sij
2
To replace the hardening parameter k with σe, we
substitute Eq. (12) in Eq. (9) and rewrite it as
3
(13)
f(σ ij ,εijp ,k)= sij sij -σ e2 (ε p )=0
2
From the definition of the associated flow rule, g=f,
the derivatives of g and f are found as
f
g
(14)
=
=3sij
σ ij σ ij
The strain history for the material is the record of the
length of the effective plastic strain path
dσ e
(20)
ε p =  dε = 
p H (σ )
p
e
d
function have to be related to the experimental
uniaxial stress-strain curve. To this end, the stress
variable σe, called the effective stress, and the strain
variable εp, called the effective strain, are introduced.
Since the effective stress should reduce to the stress σl
in the uniaxial test, it follows that the loading function
F(σij) can be expressed as F(σ ij )=Cσ en . For a von
Mises material, F(σij ,εijp )=J 2 , then
J 2 =Cσen
(11)
in press
with
h=4  3G+H p  σ e2
(21)
and
(16)
un
where the second order tensor Hkl associated with the
yield function, f, is defined as
f
(17)
H kl 
Cijkl  6Gskl
σ ij
The parameter Hp is a plastic modulus associated
with the rate of expansion of the loading surface, and
it can be defined as the slope of the uniaxial
stress-plastic strain curve at the current value of σe.
dσ
(18)
Hp  e
d p
For the F(J2,J3) material, such as the von Mises
material, which is pressure independent when plastic
flow occurs, the effective plastic strain increment is
defined as
2 p p
(19)
d p 
d ij d ij
3
5
1


=  Cijkl  H ij H kl  dεkl
h


2


36G
=  Cijkl 
sij skl  dεkl
h


ed
Then Eq. (5) becomes dεijp =dλ(3sij ) , where dλ is a
scalar function to be determined by the consistency
condition df=0 as
1 f
1
(15)
dλ=
Cijkl dεkl = H kl dεkl
h σ ij
h
1


=  Cijst  H st H ij  dεst
h


Hence
ep
p
dσ ij =Cijkl
d  kl =(Cijkl +Cijkl
)d  kl
ep
p
Cijkl
=Cijkl +Cijkl
(22)
with
1
36G 2
p
Cijkl
=  H ij H kl = 
sij skl
(23)
h
h
In conclusion, the complete stress-strain
relationships for an elastic-plastic work-hardening
material can be expressed as follows:
(i) When f(σ ij ,εijp ,k)=0 with dλ>0, the material is in
ep
dεkl and
a plastic loading state. We have dσ ij =Cijkl
ep
Cijkl is given in Eq. (22) and Eq. (23).
(ii) When f(σ ij ,εijp ,k)<0, or f(σ ij ,εijp ,k)=0 with dλ≤0,
the material is in an unloading or neutral loading state.
We have dσ ij =Cijkl dεkl and Cijkl is given in Eq. (3).
4. Finite Element Analysis
d
material properties, will also result in FE calculations
becoming more difficult to converge. Therefore, in
this paper, we incorporate the above-mentioned
simplified constitutive laws with the following
adequate FE model, and thus expect to maintain
accuracy between numerical and experimental results
at any deformation stage until reaching the
experimental ultimate fracture strain, without
aborting due to numerical errors or local distortions
occurring in the model.
un
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Calculations were carried out by means of the FE
method in this study, and the results were verified by
the uniaxial tensile experiments conducted by Sasaki
et al. (2001) on 3Y-TZP, 2Ge-1Mg, 2Ge-1Ca, and
2Ge-2Ti superplastic ceramics. The FE simulation
was performed using a commercial FE package,
Abaqus (Dassault Systèmes Corporation, 2014). The
elastic-plastic constitutive model used in this study
was an Abaqus built-in metal plasticity material
model, in which the isotropic hardening rule could be
included. The experimental constitutive stress-strain
hardening data could be defined in tabular form as the
input for the metal plasticity material model in
Abaqus. When the analysis begins, Abaqus connects
the data points with piecewise linear line segments to
approximate the nonlinear stress-strain relationships
of materials. It is unnecessary to input all of the
experimental stress-strain data because if a simplified
material constitutive law is used in an FEM, which
can simulate the major stages of deformation without
losing the accuracy between the FEA and the
experimental results, then it can be regarded as a
practical one for the purpose of applications.
Therefore, in this study, a simplified constitutive law
was created based on the following rule: the turning
points for different deformation stages of the
experimental stress-strain curves have piecewise
linear connections. According to this rule, there were
three to four data points chosen for every studied
composition, and the simplified constitutive laws
used in Abaqus for the FE simulations are shown in
Fig. 3. Although these input data seem to simplify by
following the complete experimental stress-strain
characteristics, the FEA of superplastic deformation
problems may be aborted before reaching the
experimental ultimate fracture strain as it will lead to
numerical errors for various reasons. For instance, (1)
extreme superplastic deformation will lead to an
increase of element aspect ratios which usually are
accompanied by cumulative numerical errors and
distortion of the mesh; (2) suddenly changed material
properties or geometric shapes at some regions
exhibiting stress concentrations may result in singular
points, which will increase the difficulty of numerical
convergence or lead to severe local deformation in a
few elements; (3) strain-hardening or strain-softening
in press
6
Fig. 3 The simplified constitutive laws of 3Y-TZP and three
co-doped materials
Fig. 4 Geometry of the finite element model
It should be noted that the curves in Fig. 3 are
recorded as true stresses versus nominal strains (or
engineering strains) from experimental data. In order
to simulate the large deformation of the superplastic
materials for finite element analysis, those true
stress-nominal strain curves are transformed into true
stress-true strain curves and input to the Abaqus
program. Let εnom be the nominal strain and εtrue be the
true strain. The transformation equation between
these two strains can be written as:
εtrue =ln(1+εnom )
(24)
The cases studied here addressed the problem of a
homogeneous and isotropic material experiencing
uniform uniaxial loading on its uniform cross section.
According to the theory of continuum mechanics, the
internal axial stresses and strains existing in elements
throughout the model should be uniformly distributed;
thus, there is no need to perform a convergent analysis
on the element numbers of the model. However, we
observed that the aspect ratio of elements should be
less than a maximum value of 4 to avoid a numerical
error in the FE calculation. By arranging the numbers
of the elements equal to 7, 1, and 1 corresponding to
the x, y, and z directions, respectively, the aspect ratio
of the elements in the model was 1.04.
ite
d
In the present analyses, the stress concentration
caused the greatest elongation to occur at only a few
elements on the junction of the outside clamping
region and the middle gauge region of the bone
samples. In order to avoid this, the geometry of the
studied cases introduced in Section 2 is schematized
in Fig. 4, which only includes a gauge region with a
cross section of 2mm×2mm and a length of 13.4 mm.
in press
ed
Fig. 5 Boundary conditions of the finite element model
Fig. 6 Element distribution of the finite element model
un
The analyses were conducted with a
three-dimensional simulation with the advantage of
being able to clearly examine the stress and
deformation conditions in the model. For the purpose
of simulating the uniaxial tensile test with a strain
control technique, the boundary and loading
conditions are shown in Fig. 5 and are described as
follows: ux=0 is applied to the PQRS plane; uy=0 is
applied to the PQ and TU line segments; uz=0 is
applied to the P and T points; and a uniform
deformation ux=0.15 m is applied to the TUVW plane.
The element used in the analysis is the three
dimensional solid (continuum) element, C3D20,
which has 20-node with three degrees of freedom per
node (displacements in the x, y, and z directions). The
finite element mesh of the specimen is shown in Fig. 6.
7
Fig. 7 Comparison between the FEA and experimental
results for the stress-strain characteristics of 3Y-TZP and
three co-doped materials
Fig. 8 Final deformed shapes in the FEA for 3Y-TZP and
three co-doped compositions
5. Results and Discussion
The comparison of true stress vs. nominal strain
d
the experimental samples agree well with the results
of the FE simulation. These results show that the
presented FEM is a suitable method by which to
simulate the mechanical behavior of superplastic
co-doped 3Y-TZP ceramics.
ite
relationships between the experimental results and
numerical analysis for the four studied compositions
is presented in Fig. 7. Regardless of whether the
brittle 3Y-TZP or the comparatively superplastic
2Ge-1Mg, 2Ge-1Ca, and 2Ge-2Ti ceramics were
examined, this figure shows that the experimental
stress-strain relationships were in good agreement
with the FEA results for any stage of deformation:
elastic loading, horizontal plastic stretch, strain
hardening, and strain softening until ultimate fracture.
The original undeformed and the final deformed
shapes of the FEA for the four compositions are
depicted in Fig. 8.
in press
ed
Fig. 11 Stress and strain history of the FEA at different
stages of deformation for 2Ge-1Ca
stages of deformation for 3Y-TZP
un
stages of deformation for 2Ge-2Ti
stages of deformation for 2Ge-1Mg
By comparing Fig. 8 with the corresponding
experimental results in Fig. 2, the errors of the
maximum nominal strain between the FEA and
tensile tests can be calculated as: 3Y-TZP, 0.26%;
2Ge-1Mg, 0.46%; 2Ge-1Ca, 0.64%; and 2Ge-2Ti,
0.93%; all of them are small enough to obtain
accuracy. Moreover, the final deformed shapes,
length, width, and thickness of the gauge section in
8
The axial true stress contour defined by different
colors and the values of nominal strain at different
stages of deformation for 3Y-TZP, 2Ge-1Mg,
2Ge-1Ca, and 2Ge-2Ti compositions are shown in
Figs. 9 to 12, respectively, and they represent the
deformation stages from the beginning of the analysis
to the end. In these figures, the grade of axial true
stress contour in pascal (Pa) is represented by the
different colors at the lower left corner. Using Fig. 12,
for example, to explain other features, there are five
samples in the figure, and the undeformed sample on
the top indicates the initial condition of the analysis,
in which the axial nominal strain and the axial true
stress are both equal to zero, and the zero stress is
located on the blue color contour range of -5.255e+2
to 1.344e+6 Pa. The second, green, sample from the
top of Fig. 12 indicates that the analysis is in the stage
d
simulations in Abaqus is based on piecewise linear
connections at the turning points of different
deformation stages on the experimental stress-strain
curves. The presented FEM was verified with the
tensile test experiments on superplastic 3Y-TZP,
2Ge-1Ca, 2Ge-1Mg, and 2Ge-2Ti ceramics
performed by Sasaki et al. (2001).
During the analysis, in order to avoid the
analysis aborting before reaching the experimental
ultimate strain due to stress concentration and also to
avoid an unreasonable peak elongation occurring at
only a few elements on the junction of the outside
clamping region and the middle gauge region of the
bone samples, a geometry was constructed in the
numerical FEM that only included the gauge region.
The results showed that the stress-strain relationships
analyzed by the presented FEM agreed well with the
experimental data, and the errors for the maximum
stress and strain were all less than 1% for the four
compositions studied. Furthermore, the final
deformed shape (i.e. width and thickness) of the finite
element analysis were consistent with the results of
tensile tests. These verifications confirm the
reliability of the presented FEM, which can be used to
analyze the mechanical behavior of materials such as
superplastic co-doped 3Y-TZP ceramics. This paper
gives a feasible model for simulating the constitutive
characteristics of superplastic ceramics and to some
degree makes up for the lack of numerical analysis
available for this kind of material. In addition, this
paper offers a model for analyzing applications
related to the development of manufacturing process
improvements, mechanical analyses, fracture
predictions, and safe design specifications for
superplastic ceramics.
un
ed
ite
of plastic stretch with an axial nominal strain of
127.98%, and the axial true stress of 5.5 MPa can be
determined by corresponding to the 2Ge-2Ti
stress-strain relationships for the FEA results shown
in Fig. 7, and the stress locates at a green color
contour range of 5.376e+6 to 6.720e+6 Pa. Similarly,
the third green, the fourth red, and the fifth blue
samples from the top of Fig. 12 reveal that the
specimens are experiencing corresponding strain
hardening deformation condition (axial true stress =
7.75 MPa, and axial nominal strain = 570.76%), the
maximum ultimate stress state (axial true stress =
13.44 MPa, and axial nominal strain = 984.14%), and
the ultimate fracture strain state (axial true stress = 0
MPa, and axial nominal strain = 997.18%),
respectively., One can interpret Figs. 9, 10, and 11 in
the same way corresponding to the results of 3Y-TZP,
2Ge-1Mg, and 2Ge-1Ca, respectively. The four
figures (Figs. 9 to 12) reveal that the axial stresses and
strains are distributed uniformly in the model at every
deformation stage. The reason for this was mentioned
in Section 4 in which we described the problem type
analyzed in the FEM as a uniform uniaxial loading
acted on a uniform cross section of homogeneous and
isotropic materials. Hence, the studied models
became uniformly longer and thinner until the
analysis stopped at a strain near the experimental
ultimate fracture strain, and the axial stress varied
with the axial strain, as shown in the relationships
illustrated in Fig. 7. These results confirmed the
reliability of the presented FEM.
in press
6. Conclusions
This paper presented a numerical finite element
model (FEM) based on the theory of plasticity to
simulate the uniaxial stress-strain progress of
co-doped 3-mol% Y2O3-stabilized tetragonal-zirconia
polycrystals (3Y-TZP). The material model
embedded in the FEM was an elastic-plastic model,
which simulated the elastic response using Hooke’s
law and the plastic strain hardening response using
the flow rule associated with the von-Mises yield
criterion combined with the isotropic hardening rule.
The simplified constitutive law used for the numerical
9
References
Abu-Farha, F.K., Khraisheh M.K., 2007. Analysis of
superplastic deformation of AZ31 magnesium alloy.
Advanced Engineering Materials, 9(9):777-783.
Bieler, T.R., Mishra, R.S., Mukherjee, A.K., 1996.
Superplasticity in hard-to-machine materials. Annual
Review of Materials Science, 26:75-106.
Chandra, N., 1988. Analysis of superplastic metal forming by a
finite element method. International Journal for
Numerical Methods in Engineering, 26(9):1925-1944.
Chen, W.-F., Han, D.-J., 2007. Plasticity for Structural
d
Materials Processing Technology, 150(1-2):76-83.
Liew, K.M., Tan, H., Tan, M.J., 2003. Finite element modeling
of superplastic sheet metal forming for cavity sensitive
materials. Journal of Engineering Materials and
Technology, 125(3):256-259.
Lin, J., 2003. Selection of material models for predicting
necking in superplastic forming. International Journal of
Plasticity, 19(4):469-481.
Nazzal, M., Abu-Farha, F., Curtis, R., 2011. Finite element
simulations for investigating the effects of specimen
geometry in superplastic tensile tests. Journal of
Materials Engineering and Performance, 20(6):865-876.
Nazzal, M.A., Khraisheh, M.K., 2008. Impact of Selective
Grain Refinement on Superplastic Deformation: Finite
Element Analysis. Journal of Materials Engineering and
Performance, 17(2):163-167.
Nazzal, M.A., Khraisheh, M.K., Darras, B.M., 2004. Finite
element modeling and optimization of superplastic
forming using variable strain rate approach. Journal of
Materials Engineering and Performance, 13(6):691-699.
Sasaki, K., Nakano, M., Mimurada, J., Ikuhara, Y., Sakuma, T.,
2001. Strain hardening in superplastic co-doped
Yttria-stabilized tetragonal-zirconia polycrystals. Journal
of the American Ceramic Society, 84(12):2981-2986.
Tao, J., Keavey, M.A., 2004. Finite element simulation for
superplastic forming using a non-Newtonian viscous
thick section element. Journal of Materials Processing
Technology, 147(1):111-120.
Wakai, F., Kondo, N., Shinoda, Y., 1999. Ceramics
superplasticity. Current Opinion in Solid State and
Materials Science, 4(5):461-465.
Wakai, F., Sakaguchi, S., Matsuno, Y., 1986. Superplasticity of
Yttria-stabilized tetragonal ZrO2 polycrystals. Advanced
Ceramic Materials, 1:259-263.
Wang, N.M., Budiansky, B., 1978. Analysis of sheet metal
stamping by a finite-element method. Journal of Applied
Mechanics, 45(1):73-82.
Yenihayat, O.F., Mimaroglu, A., H. Unal, A.H., 2005.
Modelling and tracing the super plastic deformation
process of 7075aluminium alloy sheet: use of finite
element technique. Materials & Design, 26(1):73-78.
Zienkiewicz, O.C., Godbole, P.N., 1974. Flow of plastic and
visco-plastic solids with special reference to extrusion
and forming processes. International Journal for
Numerical Methods in Engineering, 8(1):1-16.
un
ed
ite
Engineers, J. Ross Publishing.
Chen, Y., Kibble, K., Hall, R., Huang, X., 2001. Numerical
analysis of superplastic blow forming of Ti–6Al–4V
alloys. Materials & Design, 22(8):679-685.
Dassault Systèmes Corporation, SIMULIA Abaqus Analysis
User's Manuals, Theory Manuals and Example Problems
Manuals, Version 6.14, France, 2014.
Ding, X.D., Zbib, H.M., Hamilton, C.H., A. E. Bayoumi, A.E.,
1995. On the optimization of superplastic blow-forming
processes. Journal of Materials Engineering and
Performance, 4(4):474-485.
Domínguez-Rodríguez, A., Gómez-García, D., 2010.
Superplasticity in ceramics: applications and new trends.
Key Engineering Materials, 423:3-13.
El-Morsy, A., Akkus, N., Manabe, K., Nishimura, H., 2001.
Evaluation of superplastic material characteristics using
multi-dome forming test. Materials Science Forum,
357-359:587-592.
Garvie R.C., Hannink, R.H., Pascoe, R.T., 1975. Ceramic steel?
Nature, 258(5537):703-704.
Giuliano, G., 2005. Simulation of instability during
superplastic deformation using finite element method.
Materials & Design, 26(4):373-376.
Giuliano, G., 2006. Failure analysis in superplastic materials,”
International Journal of Machine Tools and Manufacture,
46(12-13):1604-1609.
Hassan, N.M., Younan, M.Y., Salem, H.G., 2003. The
finite-element deformation modeling of superplastic
Al-8090. JOM, 55(10):38-42.
Hsu, T.-C., Chu, C.-H., 1995. A finite-element analysis of
sheet metal forming processes. Journal of Materials
Processing Technology, 54(1-4):70-75.
Hsu, T.-C., Shien, I.-R., 1997. Finite element modeling of
sheet forming process with bending effects. Journal of
Huh, H., Han, S.S., J. S. Lee, J.S., Hong, S.S., 1995.
Experimental verification of superplastic sheet-metal
forming analysis by the finite-element method. Journal of
Jiménez-Melendo, M., Domínguez-Rodríguez, A., A.
Bravo-León, A., 1998. Superplastic flow of fine-grained
Yttria-stabilized zirconia polycrystals: constitutive
equation and deformation mechanisms. Journal of the
American Ceramic Society, 81(11):2761-2776,
Jiménez-Melendo, M., and Dominguez-Rodríguez, A., 1999.
Like-metal
superplasticity
of
fine-grained
Y2O3-stabilized zirconia ceramics. Philosophical
Magazine A, 79(7):1591-1608.
Kim, Y.H., S. S. Hong, S.S., Lee, J.S., Wagoner, R.H., 1996.
Analysis of superplastic forming processes using a
finite-element method. Journal of Materials Processing
Technology, 62(1-3):90-99.
Lee, C.H., Kobayash, S., 1973. New solutions to rigid-plastic
deformation problems using a matrix method. Journal of
Manufacturing Science and Engineering, 95(3):865-873.
Li, G.Y., Tan, M.J., Liew, K.M., 2004. The finite-element
deformation modeling of superplastic Al-8090. Journal of
in press
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